When I receive a bank statement, I review all the transactions. The problem is my failing memory. I do not remember when and where I last took money from an ATM, and how much. So I decided to create a pattern. I used to take cash in multiples of 100. Probably most people do that. A better idea is to always take the amount that has a fixed remainder modulo 100, but not 0. For example, let’s say I take one of the following amounts: 40, 140, 240, or 340. Sometimes I need more money, sometimes less. This set of numbers covers all of my potential situations, but my pattern is that all the numbers end in 40. This way if someone else gets access to my account, they will almost surely take a multiple of 100. I will be able to discover a fraud without remembering the details of my last withdrawal.

In addition, when I first started doing this, I was hoping I wouldn’t need to wait until I review my own statement to discover problems. My hope was that if a thief tried to take cash from my account, my bank would notice a change in the pattern and notify me immediately. Now I realize that this was wishful thinking. I doubt that banks are as smart as I am.

One day I should try an experiment. I should go to an ATM I never use and withdraw 200 dollars. I wonder if my bank would notice.

A long time ago, before anyone ever heard about ultrasound, there was a psychic who could predict the gender of a future child. No one ever filed a complaint against him.

The psychic had a journal in which he wrote the client’s name and the gender of the future child. The beauty of the scam was that what he wrote in the journal was the opposite gender that he had predicted. Whenever a client complained that the gender was wrong, he would show the journal and argue that the client had misunderstood.

Happy clients don’t return to complain.

Oh, the power of conditional probability! It is useful to understand it to run scams or to expose them.

I recently posted a set of problems from a Russian magazine for middle school children. Now it’s time for solutions.

Problem 1. There are 6 glasses on the table in a row. The first three are empty, and the last three are filled with water. How can you make it so that the empty and full glasses alternate, if you are allowed to touch only one of the glasses? (You can’t push one glass with another.)Solution. Pour the water from the fifth glass to the second glass.

Problem 2. If it is raining at midnight, with what probability will there be sunshine in 144 hours?Solution. In 144 hours it be midnight. Assuming we are not close to the earth’s poles, there will be no sunshine.

Problem 3. How can you fill a cylindrical pan exactly half-full of water?Solution. Fill it with water, then tilt so that the water slowly runs out until you are about to see the rim of the floor of the pan.

Problem 4. The Jackal always lies; the Lion always tells the truth. The Parrot repeats the previous answer—unless he is the first to answer, in which case he babbles randomly. The Giraffe replies truthfully, but to the previous question directed to him: his first answer he chooses randomly.
The Wise Hedgehog in the fog stumbled upon the Jackal, the Lion, the Parrot, and the Giraffe, although the fog prevented him from seeing them clearly. He decided to figure out the order in which they were standing. After he asked everyone in order, “Are you the Jackal?” he was only able to figure out where the Giraffe was. After that he asked everyone, “Are you the Giraffe?” in the same order, and figured out where the Jackal was. But he still didn’t have the full picture. He started the next round of questions, asking everyone, “Are you the Parrot?” After the first one answered “Yes”, the Hedgehog understood the order. What is the order?Solution. On the question “Are you the Jackal?” the Jackal and the Lion have to answer no. The Giraffe has to say “yes.” The Parrot could not have said “yes,” because in this case the Hedgehog, knowing where Giraffe is, could have figured out where the Parrot is. That means, in the first round there was only one “yes.” We also can conclude that the Parrot is not after the Giraffe. On the question “Are you the Giraffe?” the Lion would have said “no,” and the Jackal would have said “yes.” The argument here is similar to the previous discussion: the Parrot couldn’t have said “yes.” We also know that the Parrot is not after the Jackal. So the Parrot is either the first or is after the Lion. After the Hedgehog’s last question, we know that the Lion can’t be the first. That means the Parrot is the first. Which means the previous animal said “yes,” implying that the Jackal is the last. So there are two possibilities: The Parrot, the Lion, the Giraffe, the Jackal, or the Parrot, the Giraffe, the Lion, the Jackal. But in the second arrangement, the Hedgehog would have differentiated the Lion from the Parrot before the last question, as the Parrot couldn’t have been after the Giraffe. The answer is: the Parrot, the Lion, the Giraffe, the Jackal.

Problem 5. There are 12 cards with the statements “There is exactly one false statement to the left of me,” “There are exactly two false statements to the left of me,” …, “There are 12 false statements to the left of me.” Pete put the cards in a row from left to right in some order. What is the largest number of statements that might be true?Solution. Suppose there are more than six statements that are true. That means one of the cards with a number more than six is true, meaning that there are at least 7 false statements. This is a contradiction. Suppose there are 6 true cards. They have to be cards with numbers one through six. In addition, no pair of adjacent cards can both be true. Arranging the cards in the order 7, 1, 8, 2, 9, 3, 10, 4, 11, 5, 12, 6 works. Notice that we can permute the cards with numbers more than 6 and it will still work.

Problem 6. Olga Smirnov has exactly one brother, Mikhail, and one sister, Sveta. How many children are there in the Smirnov family?Solution. I shouldn’t have posted this problem. In Russian it is clear that Olga is a girl and the answer is 3 children. In English this problem might be confusing.

Problem 7. Every next digit of number N is strictly greater than the previous one. What is the sum of the digits of 9N?Solution. Remember that 9N = 10N − N. That means the i-th digit of 9N is the (i+1)-th digit of N minus the the i-th digit of 9N. The next to last digit of 9N is the last digit of N minus the next to last digit of N minus 1 (because of the carry). The last digit of 9N is 10 minus the last digit of N. Summing this up, the answer is 9.

Problem 8. Nine gnomes stood in the cells of a three-by-three square. The gnomes who were in neighboring cells greeted each other. Then they re-arranged themselves in the square, and greeted each other again. They did this one more time. Prove that there is at least one pair of gnomes who didn’t get a chance to greet each other.Solution. Color the square in checkerboard colors. There are 8 different ways to assign colors to gnomes for the three rounds. That means there are two gnomes who stood on the same colors all three times. They couldn’t have greeted each other.

One day I received a call on my home line. I do not like calls from strangers, but the guy knew my name. So I started talking to him. I assumed that it was some official business. He told me that their company monitors Internet activities, and that my computer is emitting viruses into the Internet traffic degrading Internet performance. All I need to do is to go to my computer and he will instruct me how to get rid of my viruses.

While he was saying all this, I covered my phone’s microphone and made a call to the police from my cell phone. I was hoping the police could trace the call and do something while I kept the line to the guy open. The police told me to hang up. They said there is nothing they can do.

Meanwhile, the guy on the phone kept directing me to my Start button while I kept telling him that I can’t find it. After talking to the police, I got so angry that I told the guy that I wasn’t actually looking for the Start button, but talking to the police. So the guy asks, “What does the police say?”

These people are laughing at us. They know that the police do nothing. And then continued instructing me about my Start button.

Recently I wrote that my blog is under attack by spam comments. Most of the comments were caught by my spam-filter Akismet, the best-known filter for WordPress. I was receiving about 50,000 comments a day and 200 of them were sneaking through this filter. I had to moderate those and delete them. This was an extreme waste of my time. But I can understand that the bots achieved some goal. None of their comments made it to my website, but at least I myself was made aware of opportunities for hair removal in Florida.

The comments crashed my server and I had to install CAPTCHAs. I was happy that the number of comments that I had to moderate went down, but the total number of comments was still so high that my server kept crashing. Now that the comments are blocked from human view, why are they still pouring in? One software package is trying to inform the other software package about weight-loss wonder drugs. I am convinced that Akismet is not interested.

My hosting provider couldn’t handle the traffic and asked me to upgrade to a more advanced hosting package. It’s annoying that I have to pay a lot of money for these bots’ attempts to sell Akismet fake Louis Vuitton bags.

The upgrade was too expensive, so I tried a different solution. I closed comments for older posts. It didn’t help. The bad software continued trying to leave comments that can’t be left. They especially like my post about Cech cycles, called A Mysterious Bracelet. My weblog tells me that every second someone downloads this page and tries to leave a comment. But no one will ever see these comments. Even Akismet will never know what it is missing: it might have had a chance to make $5,000 a day from home.

I have an idea for a start-up medical insurance company for Massachusetts. My insurance will have an infinite deductible. That means you pay your own bills. The cost of insurance can be very low, say $100 a year, as I do not need to do anything other than to send you a letter confirming that you have medical insurance. People who otherwise will be fined up to $900 for being uninsured will run in droves to buy my insurance.

I would like to thank my readers Qiaochu Yuan, Mark Rudkin, “ano” and Paul who alerted me to the problem. Viewers who were using the Google Chrome browser and who tried to visit my website got this message: “This site contains content from howmanyoffers.com, a site known to distribute malware.”

It took me some time to figure out what was going on. It appears that on June 19 someone from 89-76-135-50.dynamic.chello.pl hacked into my hosting account and added a script to all my html files and to my blog header. It seems that the script was dormant and wasn’t yet doing bad things.

As soon as I grasped what was going on, I replaced all the affected files.

I have had my website for many years without changing my hosting password. Unfortunately, passwords, not dissimilar to humans, have this annoying tendency to become weaker with age. I wasn’t paying attention to the declining strength of my password and so I was punished.

Now I have fixed the website and my new password is: qwP35q2054uWiedfj052!@#$%.

I am sitting in front of my computer and scheming, or, more precisely, scamming. I am inventing scams as a way of raising awareness of how probability theory can be used for deception.

My first scam is my lottery project. Suppose I create and run a private lottery. I will award minor payments to some participants, while promising a grand prize of one hundred million dollars. However, there will be a very small probability that anyone will win the big payout. My plan is to live lavishly on my proceeds, hoping no one ever wins the big ticket.

The beauty of this scheme is that nobody will complain until someone scores the top prize. After all, everyone has been receiving what I promised, and no one realizes my fraud. If nobody wins the big award until I retire, I will have built my life style on deception without having been caught.

Suppose someone wins the hundred million dollars. Oops. I am in big trouble. On the other hand, maybe I can avoid jail time. I could tell the winner that the money is gone and if s/he complains to the police, I will declare bankruptcy and we will all lose. Alternatively, I can suggest a settlement in exchange for silence. For example, we could share future proceeds. Probability theory will help me run this lottery with only a small chance of being exposed.

But even a small chance of failure will cause me too much stress, so I have come up with an idea for another scam. I will write some complicated mathematical formulas with which to persuade everyone that global warming will necessarily produce earthquakes in Boston in the near future. Then I’ll open an insurance company and insure everyone against earthquakes. As I really do not expect earthquakes in my lifetime, I can spend the money. I’ll just need to keep everyone scared about earthquakes. This time I can be sure that I won’t be caught as no one will have a reason to complain. The only danger is that someone will check my formulas and prove that I used mathematics to lie.

Perhaps I need a scam that covers up the lie better. Instead of inventing an impossible catastrophe, I need to insure against a real but rare event. Think Katrina. I collect the money and put aside money for payouts and pocket the rest. But I actually tweak my formulas and put aside less than I should, boosting my bank account. I will be wealthy for many years, until this event happens. I might die rich but if this catastrophe happens while I’m still alive, I’ll declare bankruptcy.

Though I was lying to everyone, I might be able to avoid jail time. I might be able to prove that it was an honest mistake. Mathematical models include some subjective parameters; besides, everyone believes that nature is unpredictable. Who would ever know that I rigged my formulas in my favor? I can claim that the theory ended up being more optimistic than reality is. Who could punish me for optimism?

Maybe I can be accused of lying if someone proves that I knew that the optimistic model doesn’t quite match the reality. But it is very difficult for the courts to punish a person for a math mistake.

When I started writing this essay, I wanted to write about the financial crisis of 2008. I ended up inventing scams. In a way, I did write about the financial crisis. My scams are simplified versions of what banks and hedge funds did to us. Will we ever see someone punished?

I once wrote a story about a mistake that my medical insurance CIGNA made. They had a typo in the year of the end date of my insurance coverage in their system. As a result of this error, they mistakenly thought they had paid my doctors after my insurance had expired and tried to get their money back. While I was trying to correct all this mess, an interesting thing happened.

To help me explain, check out the following portion of my bill. (If it looks a bit funny, it’s because I cut out some details including the doctor’s name).

On the bill you can see that I had a mammography for which I was charged $493.00, but CIGNA paid only $295.80. The remaining $197.20 was removed from the bill as an adjustment, as frequently happens because of certain agreements between doctors and insurance companies. A year later when CIGNA made their mistake, they requested that the payment be returned. You can see on the bill that once the payment was reversed, my doctors reversed the adjustment too.

When CIGNA fixed the typo, they repaid the doctors, but the adjustment stayed on the bill, which the doctors then wanted me to pay. And that was only one of many such bills. It took me a year of phone calls to get the adjustments taken off, but this is not what I am writing about today.

If not for this mistake, I would have never seen these bills and the revealing information on the different amounts doctors charge to different parties, and how much they really expect to receive. As you can see my doctors wanted 67% more for my mammogram than they later agreed to.

The difference in numbers for my blood test was even more impressive. I was charged $173.00, and the insurance company paid $30.28 — almost six times less.

If I ever need a doctor and I don’t have insurance, I will take these bills with me to support my request for a discount. I do not mind if you use this article for the same purpose.

I do not recall your name and I’m not sure where you got my email address from, but I really appreciate you contacting me. I am excited by your Two-Procedures-For-The-Price-Of-One offer. I am really looking forward to my enlarged penis and my DDD breasts.

Meanwhile, I can give you a unique group discount on IQ tests. I can test the IQ of all your company employees for the price of one test. Moreover, you do not need to waste even a minute. Actually, no one even needs to answer any questions. You can send me your $500 check to the address below and I will promptly send you the IQ report, the accuracy of which I can guarantee.